After several characters up to Level 65, here is one of the way you can use to max your character's level. It's not the fastest, because I hate grind.

From Level 60, just go with the new Story Quests + Vanguard Quests. If you're a damager class, you can just solo all quest. But if you're a healer, either Mystic or Priest, I suggest you party with another damager, just to speed up the process.

If you finish Story Quests + Vanguard Quests excluding Dungeon Quests, your level should be around 63 and half. After doing all that, get a party and do the Dungeon Quest.

If you hate grind the new dungeon, just go to Battleground. You can choose between Kumasylum, Corsairs' or the new one Monkeydrome.

Make sure to do the BG's daily before the end of the day.

That's all right now. Gonna update this soon, when I'm at home.